{"title":"Some applications of Gröbner-Shirshov bases to Lie algebras","authors":"Luis Mendonça","doi":"10.1016/j.jpaa.2024.107773","DOIUrl":null,"url":null,"abstract":"<div><p>We show that if a countably generated Lie algebra <em>H</em> does not contain isomorphic copies of certain finite-dimensional nilpotent Lie algebras <em>A</em> and <em>B</em> (satisfying some mild conditions), then <em>H</em> embeds into a quotient of <span><math><mi>A</mi><mo>⁎</mo><mi>B</mi></math></span> that is at the same time hopfian and cohopfian. This is a Lie algebraic version of an embedding theorem proved by C. Miller and P. Schupp for groups. We also prove that any finitely presentable Lie algebra is the quotient of a finitely presented, centerless, residually nilpotent and SQ-universal Lie algebra of cohomological dimension at most 2 by an ideal that can be generated by two elements as a Lie subalgebra. This is reminiscent of the Rips construction in group theory. In both results we use the theory of Gröbner-Shirshov bases.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022404924001701","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We show that if a countably generated Lie algebra H does not contain isomorphic copies of certain finite-dimensional nilpotent Lie algebras A and B (satisfying some mild conditions), then H embeds into a quotient of that is at the same time hopfian and cohopfian. This is a Lie algebraic version of an embedding theorem proved by C. Miller and P. Schupp for groups. We also prove that any finitely presentable Lie algebra is the quotient of a finitely presented, centerless, residually nilpotent and SQ-universal Lie algebra of cohomological dimension at most 2 by an ideal that can be generated by two elements as a Lie subalgebra. This is reminiscent of the Rips construction in group theory. In both results we use the theory of Gröbner-Shirshov bases.